betting strategies

In game-theoretic statistics we have some wealth process ${\mathcal{K}}_t={\prod }_{i\le t}{S}_i$ where the “payoffs” $S_i$ are e-values depending on some predictable “bet” ${\lambda }_i$. The e-values might be of exponential form (see exponential inequalities) or of the form $1+{\lambda }_i{G}_i$ (eg estimating means by betting).

We want ${\mathcal{K}}_t$ to increase if some null is not true. If a particular alternative is known, we might consider designing the $S_i$ according to some growth rate conditions in sequential testing, if they can be solved exactly. If not, or if a particular alternative is not known, we need to do something else. Here we gather various strategies.

GRAPA and aGRAPA

Idea behind GRAPA is to maximize log-wealth. If we don’t know a particular alternative, we can ask, at any time $t$, what would have maximized the log-wealth in hindsight? GRAPA and aGRAPA are approximations to this answer (which does not have a closed-form solution).

LBOW

Here we aim to maximize a specific lower bound on the wealth, similar to the ELBO in variational inference. A lower bound on the wealth is usually derived using some exponential inequalities.

ONS

We may also borrow techniques from the online learning literature, such as online Newton step.